The present status on the existence and structure of static and stationary solutions of the Einstein-Vlasov system is reviewed. The structure of spherically symmetric static solutions is quite well understood and a number of their features will be discussed. I will in particular present a recent result on the existence of static massless solutions. These solutions are models of self-gravitating photon shells, or spherically symmetric geons. For axially symmetric solutions much less is known but an existence result will be described. Furthermore, I will present results from a recent numerical study, where in particular stationary toroidal solutions are found which contain ergoregions. An open question, originally posed by Wheeler, is whether or not there exist toroidal geons. This question is under investigation and I will discuss some preliminary results.